Testing the L picture and IFS fractal generation Maple 13 commands

Transcription

1 Testing the L picture and IFS fractal generation Maple 13 commands In order to make fractals with iterated function sstems it is nice to have a test procedure to make sure ou have picked our affine maps correctl (and to help ou adjust them later if necessar.) The procedure TESTMAP below, takes a list of affine functions as its input, and the result is a mapping L picture like the ones in the fractal blueprints in the director edu/~korevaar/fractals. Before tring to generate fractal eamples ou should load the subroutines in this file, b placing the cursor in an command fields ou will need, and pressing the <enter> (<return>) ke. Alternatel, ou can choose Edit/Eecute/Worksheet from the menu toolbar, and commands will be entered in document order. restart:with(plots):digits:=4: #ever time ou start a new fractal save and close our #previous fractals work. Then return to a fresh cop of this document #and restart; otherwise ou ma overwhelm #Maple and crash! Also, save frequentl. TESTMAP:=proc(funclist) #this procedure lets ou test a list of #functions in our iterated function sstem local num,#the number of functions i, #dumm inde F, #current function in list S, #corners of unit square L, #corners of letter L Sq, #unit square Llet, #letter L AS, #transf of square corners ASq,#transf of square AL, #transf of L corners ALlet, #transf of letter L Pics; #a list of pictures S:=[[0,0],[0,1],[1,1],[1,0]]; L:=[[.1,.9],[.1,.65],[.2,.65],[.2,.675],[.125,.675],[.125,.9]]: Sq:=polgonplot(S,transparenc=.9): #polgonplot connects the dots! Llet:=polgonplot(L,transparenc=1): displa({sq,llet}); num:=nops(funclist): for i from 1 to num do F:=funclist[i]: #select ith map

2 AS[i]:=map(F,S): AL[i]:=map(F,L): ASq[i]:=polgonplot(AS[i],transparenc=.9): ALlet[i]:=polgonplot(AL[i],transparenc=.9): #a plot of the transformed square and letter: Pics[i]:=displa({ASq[i],ALlet[i]}): #finall, displa the unit square and all its images: displa({sq, seq(pics[i],i=1..num)},scaling=constrained, title= fractal template ); Here is the standard affine map, which encodes AFFINE1 = a b c d e f or, in the equivalent matri representation: AFFINE1 = a c b d e f AFFINE1:=proc(X,a,b,c,d,e,f) RETURN(evalf([a*X[1]+c*X[2]+e, b*x[1]+d*x[2]+f])); Eample 1: Encode the affine transformations which epress (an isosceles) Sierpinski s triangle as a union of shrunken images: f1:=p >AFFINE1(P,.5,0,0,.5,0,0); #shrink b.5 and don t translate f2:=p >AFFINE1(P,.5,0,0,.5,.5,0); #same shrink, and translate 0.5 to the right f3:=p >AFFINE1(P,.5,0,0,.5,.25,.5); #shrink, then displace b [.25,.5] TESTMAP([f1,f2,f3]); #Check our transformations!! Since the template is correct, we ma proceed with the iteration. Start with a single point. If ou have "n" contraction mappings, then each iteration multiplies the number of points in our current set b the number "n". Your number "k" of iterations should keep the total number of points n^k under 300, 000, or Maple ma stall out on ou. S:={[0,0]}:#initial set consisting of one point 3^9; #good to keep point numbers well below 300,000, #so as not to strain Maple s memor Based on the computation above I will do nine iterations below:

3 for i from 1 to 9 do S1:=map(f1,S); S2:=map(f2,S); S3:=map(f3,S); S:= union (S1,S2,S3); pointplot(s,smbol=point,scaling=constrained, title= Sierpinski triangle ); Now if ou cop and modif this file ou can create as man contraction functions as ou want, and see what fractal the generate. Alternatel, ou can use the L picture template to reconstruct contractions for an interesting fractal ou d like to make!use these procedures to test a template and generate a fractal. Ever time ou restart ou will need to return to this window to re enter the procedures ou need. Unfortunatel the "point" resolution size in Maple 13 is prett large, so pictures tend to overstate the actual fractal. (Maple 8 was actuall better in this regard.) Eample 2: From the book "Fractals Endlessl repeated geometric figures", b Hans Lauwerier, page 95. This sstem generates a bush like fractal, with onl two generating functions! f1:=p >AFFINE1(P,.6,.6,.6,.6,0,0); #A rotation (b Pi/4), and rescaling f2:=p >AFFINE1(P,.53,0,0,.53,1,0); #pure scaling b.53, with translation TESTMAP([f1,f2]); 2 16 ; S:={[0,0]}: for i from 1 to 16 do S1:=map(f1,S): S2:=map(f2,S): S:= union (S1,S2): pointplot(s,smbol=point,scaling=constrained, aes=none, title= Figure 5.16 page 95 Lauwerier ); Printing: The resulting fractal pictures can be printed or eported as.jpg (or other) files to other documents, using menu options at the top of the Maple 13 Window. In the Math Department, there seem to be issues in printing fractal pictures with a lot of points, such as those above. One wa around that is to click on the fractal image, so that the "Plot" menu item at the top of the Maple window becomes active. You can then eport the fractal picture as a.jpg file, using the "eport" button at the bottom of the "Plot" options. You can open the.jpg file from our browser, and then print from there.

4 Testing for contractions: In order for the set map iterations to have a limit fractal, each of the transformations must be a contraction. Precisel, there must be some number strictl less than one so that distances between image points are alwas no more than this number times the corresponding distances between the domain points. Here is a procedure which will test a transformation to see if it is a contraction, in case ou re not sure. It relies on the spectral theorem for smmetric matrices, and is related to singular value decomposition for transformations. with(linalg): #old linear algebra librar STRETCHTEST:=proc(fcn) #test affine map for contraction local Atemp, #matri of affine transformation temp; #hold eigenvector information of the transpose #of Atemp times Atemp the square root of #the largest eigenvalue #is the largest stretch factor #ou want this to be less than one Atemp:=transpose (matri([fcn([1.,0]) fcn([0,0]),fcn([0,1]) fcn([0,0])])); temp:=eigenvectors(transpose(atemp)&*atemp); if (temp[2]=2 and temp[1]>=1) #uniform stretch, factor >=1 then return(print("epands uniforml, b ", temp[1], "in all directions not contraction!")); if (temp[2]=2 and temp[1]<1) #uniform stretch, factor <1 then return(print("contracts uniforml, b ", temp[1], "in all directions contraction!")); if (temp[1][1]>temp[2][1] and temp[1][1]>=1) then return(print ("not a Euclidean contraction: maimum stretch factor", sqrt(temp[1][1]), "in direction", temp[1][3] [1] )); if (temp[1][1]<temp[2][1] and temp[2][1]>=1) then returnt(print ("not a Euclidean contraction: maimum stretch factor", sqrt(temp[2][1]), "in direction", temp[2][3] [1] )); if (temp[1][1]<1 and temp[2][1]<1) then print ("contraction!");

5 Eample: f1:=p >AFFINE1(P,.7,.8,.8,.7,.6,3); f2:=p >AFFINE1(P,.5,.6,.2,.3,.8,.5); f3:=p >AFFINE1(P,1.5,0,1,.5,.25,.5); f4:=p >AFFINE1(P,.7,.2,.2,.7,.6,3); TESTMAP([f1,f2,f3,f4]); STRETCHTEST(f1); STRETCHTEST(f2); STRETCHTEST(f3); STRETCHTEST(f4);